I am curious about how to calculate the expected number of games until a gambler with $B$ dollars gets to $M>B$ dollars, or gets ruined.
I am also curious how to calculate the probability of ruin. I have seen this calculation done with a player who can win 1 dollar or lose 1 dollar. Is there a way to do this when there are $N$ different win amounts, with $N$ different probabilities? I have seen a reference to Guy Katriel's paper, but it looks at favorable games. I am looking for a derivation for unfavorable games.
Any help greatly appreciated.
The paper G. Katriel, Gambler’s Ruin: The Duration of Play, Stochastic Models 30 (2014), 251-271 http://www.tandfonline.com/doi/full/10.1080/15326349.2014.902317 considers a general unfavorable game, and gives a formula for the expected to ruin. However the case where there is also an upper bound B is not treated there. When you have an upper bound, the problem is a finite Markov chain, so at least computationally it can be treated by standard Markov chain techniques.